W,, Genetic Li = L, +

THE EFFECT OF EPISTASIS AND GAMETIC UNBALANCE ON GENETIC LOAD1

P. E. HANSCHE, S. K. JAIN AND R. W. ALLARD

Departments of Pomology, Agronomy and Genetics, Uniuersity of California, Dauis Received June 16, 1966

ENETIC load has been defined as the extent to which an equilibrium population is impaired because not all individuals are of optimum genotype. The concept, first discussed by HALDANE(1937), is the basis of theory developed by CROW(1958) and others for assessing the relative impact of various facets of the genetic system (especially mutation, segregation and inbreeding) on the average fitness of populations in genetic equilibrium. LEVENE(1 963) recently discussed various theoretical and experimental difficulties in utilizing the theory of genetic loads in the analysis of the genetic structure of populations. In its simplest form, this theory involves several assumptions among [which those of no epistasis on the fitness scale and linkage equilibrium are particularly restrictive. The purpose of this study is to investigate the effects of epistasis and gametic unbalance (linkage disequilibrium) on the manifestation of genetic loads when inbreeding is imposed on random-mating populations. Since the measures of joint inbreeding at two or more loci are influenced by gametic unbalance, the expressions developed by HALDANE(1949) for joint inbreeding are first generalized to take into account gametic and zygotic associations. The loads maintained at interacting loci are then evaluated in terms of genetic models involving several kinds of epistasis.

INBREEDING LOAD IN POPULATIONS WITHOUT EPISTASIS Genetic load has been explicitly defined, for equilibrium populations, as the fitness of the optimal genotype (by convention taken to be unity) minus the average population fitness (CROW1958). Thus, if genotypic frequencies are denoted by PI, ~23p3, . . . pn ( , 5 pi = 1 ) and the associated fitness values, 2=1 relative to the optimal value of 1, are denoted by wl,w,, w?, . . . , w,,genetic load is given by (MORTON,CROW, and MULLER1956) n L=l- ~ppiWi=l-w, i=1 when there is no epistasis. Inbreeding a population at equilibrium under random mating ordinarily increases genetic load. Assuming no interallelic interactions, MORTONet al. explicitly defined inbreeding load as Li = L, + (L- L,)f, ?'his work was supported in part by Public ITealth Sernce grant GM-10476.

Genetics 54: 1027-1040 October 196G. 1028 P. E. HANSCHE et al. where f is WRIGHT’S inbreeding coefficient (in the sense of his fixation index [WRIGHT19651 ) ,Lr is the random mating load and L, is the expected load when f = 1. The regression of load on inbreeding in such systems is linear as given by the above expression.

INBREEDING LOAD IN POPULATIONS INVOLVING EPISTASIS AND GAMETIC BALANCE Consider an infinite population and a genetic model involving two alleles at each of the loci A and B. Let the fitnesses of individual genotypes be given by w,j (Table 1).If interallelic interactions are taken into account, but the restric- tion imposed that there is “linkage equilibrium” (D= gllg33 - g13g31 = 0) , the expression for mean population fitness in random mating populations takes the form ( COCKERHAM1954) wr = giiZ.4R + glSWAb f g3lfiaR + gd3Wab > where gll, g13, g31 and g33are the gametic frequencies and WAR,ZAb, WaR and zab are the average gametic fitnesses (computed from relative zygotic fitnesses listed in Table 1). Genetic load for the random mating population thus becomes L, = 1 - wr = 1 - (g11WAR + g13wnb + g3lZaR + g33Wab). With inbreeding the average fitness for the two locus case can be expressed as Wi = Wr - f [W,+ gllg33 (ez2+ ell) - g13g3l (e12 + ~1)

TABLE 1

Gametic and zygotic frequencies and the fitness matrix for the two locus case

Female Male Marginal gametes gametes AB Ab aB ab means

g11 g13 831 g3 3 W2 2 w2 1 W12 w11 - AB g11 w~~ (g:l) (gllg13) (gllg31) (gllg3.3) w21 W2 0 w11 w10 - Ab g13 WAb k11g13) (g:,) (g13g31) (g13g33)

w12 w11 WO2 WO1 aB g31 fiae (g11g31) (g13g31) (g;,) (g31g33)

w11 w10 WO1 woo ab g33 fiab (g11gS3) (g13g33) (g.3lg3.3) (gz3)

Marginal means fiAb *nR *ab

Following COCKERHAM(1954), the epistatic parameters are: e22 = w22 - w12 - w21 i-w11 e21 = Wz1- w20 - w11+ w10 e12 = w12 - w11- WO2 + WO1 e11 = Wll - w10 - WO1 + woo EPISTASIS AND GAMETIC UNBALANCE 1029

- (gllwZ2-t g13w20 + g31wnz 4- g3wo0)I 4- F [gi1g33(ezz f ell) - g13g31 (el2 eu)1, where F is HALDANE'Sprobability of joint identity by descent for two loci (HALDANE1949) and the e,j are as defined in Table 1. By collecting terms and substituting L, for 1 - W,,L, for 1 - W,, etc., this expression can be put in the form LL = L, + (LI - Lr)f + (f - Fly, where y = glJgal(ez2 - e12- e21+ ell). The effect of epistasis on the regression of genetic load on inbreeding clearly depends upon the probability of recombination since the magnitude of F and of the gametic product g13galare both dependent upon the probability of crossing over between loci. Furthermore the effect of epistasis depends on the sign of y since (f - F) 2 0. When y is negative epistasis will decrease inbreeding load, producing a positive (upward) curvilinearity in the regression of load on inbreeding. When y is positive epistasis will increase inbreeding load, producing a negative (downward) curvilinearity in the regression of load on inbreeding.

DERIVATION OF INBREEDING LOAD FOR POPULATIONS INVOLVING EPISTASIS AND GAMETIC UNBALANCE To obtain the relationship between inbreeding and load in populations in which there is both epistasis and gametic unbalance it is first necessary to determine how gametic frequencies in the random mating population are related to zygotic frequencies after one generation of inbreeding. Then the appropriate load expression can be derived easily by multiplying zygotic frequencies by their fitness values. The relationship between gametic frequencies in random mating populations and zygotic frequencies after inbreeding: Assumptions and notation: (A) an infinitely large diploid population; (B) the probability of recombination between loci A/a and B/b is c and the probability of no recombination is (1 - c) = c'; (C) gametic frequencies are as in Table 1 (D= gllga, - g13g,,); and (D) zygotic frequencies, after random assortment of gametes with selection relaxed, are as given in Table 1. The probability that a randomly chosen zygote, x, from the population of Table 1 will produce a gamete, gl,which is AB is gll - cD. Given that the gamete gl from x is AB, then the zygote y,related to x by f and F, may produce an AB gamete, g,, as a result of any one of the following four events: 1. A and B are derived from one or ?WO common ancestors; 2. A is derived from a common ancestor and B from another source; 3. B is derived from a common ancestor and A from another source; or 4. both A and B are derived from another source. The sum of the probabilities of these four events gives the expected frequency of the zygote AABB, P(AABB),after one generation of inbreeding. The derivation of probabilities of each of the above events follows. 1. The probability of the first event, that the allelic pair AB is derived from 1030 P. E. HANSCHE et al. one or two common ancestors: The probability that the allelic pair of gamete gp is identical by descent to that of g,, given g, is AB, was shown by HALDANE(1949) to be equal to FCV.However, for the two locus case, gametic frequencies in random mating populations are, after inbreeding, not generally related to the zygotic frequencies by the function, Fw, but they are related by the functions FZ, Fti, F:: and q: (the probabilities of joint identity by descent for allelic pairs, wlth sum equal to Fxu). HALDANE'Srelationship holds only for the special case where gl and gz are statistically independent (i.e., where D = 0). This can be seen in the expression P(gl is AB n g, is A&) / P(gl is AB) = gllFIY/gll= Fxu, where the symbol n indicates joint probability and the subscript d indicates identity by descent. The expression holds, as noted above, only where the probabilities P(gl is AB) and P(gpis A&) are statistically independent. A derivation of the more generally useful functions Ft:, F$, FZ and FE: follows. The case of inbreeding by complete selfing (i.e., the case where FZ is FZ since x E VI will be considered first:

+ P(gl is AB n gz is AdBd z is AB/aB) P (z is AB/aB) + P(gl is AB n gz is A& z is AB/ab) P (z is AB/ab) + P(gl is AB flg, is AdBd z is Ab/aB) P (z is Ab/aB)

Substituting gllg33- D for g13g3,in this expression we obtain, by collecting terms and simplifying,

F4B=P(glisABP,, ng2isAdBd)=gll[(~)-6c]-(~~c2D. Consequently the probability of event (1), that A and B are derived from one or two common ancestors, given gl is AB, is P(gl is AB flg2 is AdBd)/P(gl is AB) = { gll [(%I - c'cl - (%)C2D1 / (gll -a. (1) The probabilities of joint identity by descent for the other three possible allelic pairs, Ft:, FZ: and Fg can be derived following the same probability arguments. They are:

*U--- The summation of these four probabilities is identical to the joint inbreeding function for the selfing case, F, = (%) - c'c. For sib mating the probability of joint identity by descent for the allelic pair ii, Fi:, (where z and y are sibs) is ~23XY = (%)gij(214 +2~c2+ ~2)f (%)C~D [(%) + (i/)~]. The derivation of this expression (see Appendix I) takes a different form from that of the selfing case, which ihvolves uniparental reproduction. Again Fxv = Ff + Fti + FE: + FZ;, the inbreeding function of HALDANE EPISTASIS AND GAMETIC UNBALANCE 1031 (1949) and SCHNELL(1961). It appears that F:: for any inbreeding system can be obtained by multiplying the appropriate inbreeding function, FTv,by g,, and adding (or subtracting) the term K D(Z - e2).In this term, K is the coefficient of the polynominal appropriate for F,, and 2 is the probability of joint identity by descent for the allele pair ii through one parent, when that parent is the double heterozygote. We have not, however, explored this relationship for mating systems other than selfing and sib mating. 2. The probability of the second euent, that A is deriued from a common ancestor and B is from another source, given that gl is AB can be stated explicitly as P(g1 is AB n gr is AdBs)/P(glis AB). The subscript s indicates identity in state to the homologue of gl-but not identity by descent. P(gl is AB n g2 is AdB,) = P(gl is AB n g, is AdB, 1 z is AB/AB) P(sis AB/AB) + P(gl is AB n g2 is AdB, 1 z is AB/&) P(sis AB/&) = [(%)c'c+ ('/)C'c]g11'+ [(%)C'C+ (%)c'clgllgl,=dcg,l (gll+gli). Thus, P(gl is AB n gy is AdB,)/P(gl is AB) = c'cgll (gll + gl?)/(gll- cD). (2) 3. The probability of the third euent, that B is derived from a common ancestor and A is from another source can be seen, owing to symmetry, to be P(gl is AB n g2 is A&)/P(g, is AB) = c'cgll (gll + gil)/(gll- CD) (3) 4. The probability of the fourth event, that both A and B are from another source can be stated explicitly as P(gl is AB n g, is A,B,)/P(gl is AB). P(gl is AB n gz is A$,) = P(gl is AB n gr is A$, 1 z is AB/AB)P(zis AB/AB) = r(l/)C" + (%IC2] gn2. Consequently the probability that both A and B are from another source, given gl is AB, is P(gl is AB n g2is A,B,)/P(gl is AB) = gllz [('/)c'2+(i/~cz1/(gll-c~). (4) Thus, the total probability that gl is AB and gz is AB, after one generation of complete selfing, is obtained by summing expressions (1)- (4) and multiplying by gll - cD, i.e., P(AABB) = P(gl is AB n gz is AB) = P(g2is AB I gl is AB) P(gl is AB)= [ (1) + (2) + (3)+ (4)1 [gll-cD1 = { lgll( 1/2 )-'cl - (%1 c"D) + [~'c~ll(gll+gls)l+ rc'cgll(gll+g?l)l + [(%)-YcI gn2. (51 P(glisAB) = [(l)+(2)+ (3+(4)1 Cg,,--D1=[g,l(~)-c'cl-((l/)cLD} The relationship between this general expression for P(AABB) and the special case (where D = 0) derived by HALDANEcan be clarified as follows: Substitute F for [ (%) - c'c], (f - F) for c'c (since f = %) and (1 - 2f + F)gllLfor [ (1/2) -dc] glly.Expression (5) thus becomes P(AABB) = (1- 2f + F)gl12+ (f-F) gll (gll-gdg) + fgll - ( s)c2D,where F is HALDANE'Sprobability of joint identity by descent (FXy)for complete selfing. The expected frequencies of the other nine genotypes after one generation of selfing are obtained by similar probability arguments. These frequexies, in terms of the coefficient of inbreeding, f, and the probability of joint identity by descent, 1032 P. E. HANSCHE et al. TABLE 2 Probabilities with which the 4 classes of gametes from zygote x unite with the 4 classes of gametes from zygote y, if x and y are related by f and F, and the mating system is selfing. + = I - 2f + F

P(AABB) = +gill+ (f-F)6711 k11- g33) + fg11- (%I c2D P(AABb)= 2 9 g11g13 + (f-F) (2g11g13 + gllg33 + g13g31) P(AAbb)= $ g:3 + (f-') g13 (g13 -g31) + fg13 + ($6) c2D P(AaBB) = 2 + g1lg31 + (f-F) (2g11g31 + gllg33 + g13g31) P(AB/ab) = 2 + gllg33 P(Ab/rrB)=2 + g13g31 P(Aabb) = 2 + g13g33 + (f-V(2g13gs3 + g11g33 + gi3g3i) P(aaBB) = + gil + (f4g31 k31- g13) + fg31+ (1/2)c2D P(aaBb) = 2 + g31g33 + (f-F)(2g31g3, f gllg33 + g13g31) P(aabb) = + gi3 + (f-Cg33 (g33 -g1d + fg33 - (%) CZD

Fzu,are listed in Table 2. These probabilities replace those derived by NEI (1965) who assumed relaxation of selection on zygotes during one round of random mating but did not account for the subsequent reduction in gametic unbalance in his derivation of genotypic frequencies. Derivation of genetic loud in populations maintaining gametic unbalance: The notation of COCKERHAM(1954) can be followed in assigning fitness values to each genotype (see Table 1). The average fitness of the population is obtained by multiplying genotypic frequencies by appropriate fitness values and taking the sum of the products. Thus, after one generation of selfing, Wi = W,- f [W,+ gllg33(e22+ell)- g13g31(e21+e12)- W11 + F Cgllg33 (e22+e11)-g13g31(el,+ezl) 1 - (i/~~~D(w~~-w~~-w~~+w~~). It follows that genetic load, which equals 1 - W,, is now Li = (I-W,) + f [(l-WI) - (l-Y,)l +(f-F) [y + D(e22+ell)] + (i/z)c2D(w22- WO2 - W20 + WO"), where y = g13g31 (ez2- e12- eZ1+ ell). Thus, for populations inbred by selfing, when there is epistasis and D # 0, genetic load for the 2-locus model will be: L,=L,+(L1-L,)f +(f-F) [y + D(eZ2+ell)]+(~/)c~D(w~~-~~~-w~~+zu~o~) (6) Note that the final term of expression (6), which will be designated O, measures the reduction in gametic unbalance resulting from one generation of relaxed selection. The effect of relaxing selection depends on the kind of inbreeding but not on the amount; e.g. for the sib mating case o is (i/z)cZD[(x) + (i/z)d2] x (w22 - WO2 - wzo + woo). The effect of gametic unbulance on inbreeding load.. The effect of gametic unbalance on inbreeding load can be visualized most easily by considering two kinds of epistatic systems as follows. For epistatic models in which homozygotes are of equal fitness the load expression reduces to L, = L, + (L1- L,)f + (f - F) [Iy + D(ell + eZ2)1, EPISTASIS AND GAMETIC UNBALANCE 1033 since the final term, (~/)C~D(W~~- wO2- wzo + woo)= 0. Gametic unbalance in such systems produces upward curvilinearity in the regression of load on inbreeding when D and (ell+ eZ2)are of opposite sign and downward curvilinearity when they are of the same sign. When w # 0 the load expression becomes L,=L,+ (L,--L,)f+ (f-F) [y + D (el1+e2,)I + (%)c’D(w~~-w~~-w,~,+w~~). As mentioned above, the quantity is independent of f and F. Its effect is to increase inbreeding load by a constant amount, i.e., when load is regressed on f the effect is to displace the regression line upward along the ordinate. Neither the slope nor the curvilinearity of the regression is affected. However, since the magnitude of w depends on the mating system, a spurious upward curvilinearity is imposed on the regression of load on inbreeding if load is regressed on the values off obtained from more than one mating system. Note also that this term depends on the number of generations of inbreeding since, with selection relaxed, the amount of gametic unbalance diminishes each generation. Consequently if the f’s are obtained from more than one generation of inbreeding, there will also be spurious upward curvilinearity should load be regressed on inbreeding. Some specific cases: We have chosen six epistatic models (Table 3) to illustrate the effect of epistasis and gametic unbalance on inbreeding load. These models were chosen primarily on the basis that they have all been previously proposed in explanation of experimental observations (see JAIN and ALLARD1966). The pertinent features of these models can be summarized as follows:

TABLE 3

Some specific examples of 2-locus epistasis and the values of ei

(1) Heterotic model (2) Cumulative heteiotic model .6 .8 .6 .4 .7 .3 .8 1 .o .8 .7 1 .o .6 .6 .8 .6 .5 .8 .4 (e22=e21=e12=ell=O) (e,2=e21=e12=ell=0) (3) Optimum model (4) Mixed optimum, heterotic model .6 .9 1.o .50 .80 .75 .9 1 .o .9 .80 1 .o .80 1.o .9 .6 .75 .80 .50 (e2z=e,l=e12=el,=-.2) (eZ2=el1=-. 10, e,,=e12=-.l 5) (5) Diminutive heterotic model (6) Mixed over- and underdominance .5 .9 .5 .9 .5 .9 .9 1.o .9 .5 1 .o .5 .5 .9 .5 .9 .5 .9 (e2,=e1,=--.3, e2,=e12=.3) (eZ2=ell=.9, e21=e12=-.9) (7) Mixed over- and underdominance model .8 .5 .8 .ti 1 .o .5 .7 .6 .6 (e2,=.7, eZ1=-.5, eI2=-.8, e11=.5) 1034 P. E. HANSCHE et al. (1) A heterotic model (nonepistatic) : y = 0, D = 0, e,, + ez2= 0 and wZ2- wo2- wgoi- moo = 0. Inbreeding load for this model can be expressed as Li = L, + (L,- L,)f. (2) A cumulative heterotic model: y = 0, D 2 or S 0 as a function of c and initial zygotic frequencies, ell + eZz= 0 and wZ2- woz- wzo+ who = 0. Inbreeding load for this model can be expressed as Li = L, + (L,- L,) f. (3) An optimum model: y = 0, D < 0, ez2+ el, < 0 and wZ2- woz- wzo+ woo< 0. Inbreeding load for this model for the selfing case is given by L,=Lr+ (L,-Lr)f+(f-F)D(e2,+e1,) + ( 1/2 ) c2D( W~~-WO~-W~~+WOO). (4) A mixed optimum heterotic model: y > 0, D 2 or I 0 as a function of c and initial zygotic frequencies, + e,, = 0, and wZ2- wo2- wuz0+ woo< 0. Inbreeding load for this model can be expressed (for the selfing case) as Lt = Lr + (L,-Lr)f + (f--F) y + ( 1/2)c2~(w~z-woz-~zo+~oo). (5) A diminutive heterotic model: y < 0, D 2 or 50 as a function of c and initial zygotic frequencies, ez2+ ell < 0, and wZ2- wo2- wZ0+ woo< 0. Inbreeding load can be expressed as L, = L, + (L,-Lr)f + (f-F) [ y + D(eZZ+ ell)1. (6-7) Models of mixed over-and under dominance: y > 0, D 2 or I0 as a function of c and initial zygotic frequencies, e,* f ell > 0 and wz2-wo2 - wZ0 woo= O. Inbreeding load for this model is Li = Lr + (L,--L,)f + (f-F) [ y + D(e2,+ ell)], when linkage is sufficient to maintain gametic unbalance; or Li = Lr + (L,- Lr)f + (f - F) 7, when linkage is insufficient to maintain gametic unbalance. The first step in investigating genetic load generated by the above epistatic models was to establish equilibrium gametic and zygotic frequencies. This was done by computer simulation following procedures discussed by JAIN and ALLARD (1966). The parameters D,f and L, were then determined for each of the six models (Table 4). Two breeding schemes were followed to establish inbreeding load. With scheme A the sibship families were established directly from the equilibrium random mating populations. With scheme B there was one generation of random mating with selection relaxed (all w,j = 1) to eliminate any correlation among uniting gametes (with no selection, fixation index also equals zero) before establishing the sibship families. Note that scheme A does not result in fixation index, f, equal to % and x, respectively, for half-sib and full-sib matings since, at equilibrium, the value of f is not zero for the cases studied (Table 4). In order to obtain values of f higher than those produced by a single generation of half- or full-sib matings, a mixed selfing and random mating series with the proportion of selfing, s = 0, .25, .50, .75 and 1.0 (giving f = (M)s under scheme B in the next generation) was also investigated. KEMPTHORNE(1957) has shown that f and Fw appropriate to such mixed mating systems are simple linear functions of s. Loads for breeding scheme B were computed from expression (6) and checked by computer simulation. How- ever, loads for breeding scheme A were obtained by simulation alone because the EPISTASIS AND GAMETIC UNBALANCE

Q, h h M W 0 % 0 0 9 g 9 s9 ? I I I I

3 CO m 3 m h 2 f 3 a m M c? 9 -. c? 9 9

0 h 0 m 3 0 a CO s 0 ? s -! * 3. 'I

3 h U) 8 E 3 3 "1 s s3 "7 9 8 8 K, 3 M d 2 2 3 In c? 9 c? 29 c?

00h 3 E ?i 2 (0 c? s 3 ? c?

m 3 0 m m 0 mh % 0 m % 9 9 1 c? r z

U) M M h 3 W 00h 8 3 2 M 9 !z - 9 3 9 I I I' I I I * m m W s! m 0.1 ;5 3 93 s 0 3 9 I I I I

0 0 0 0 s 9 9 m. s 9 1036 P. E. HANSCHE et al. theoretical load expressions require that f = 0 prior to the initiation of inbreeding, or that genotypic frequencies in the equilibrium population be known. The results illustrate three main features of the effect of inbreeding on genetic loads in populations where epistasis maintains gametic unbalance. First, it can be seen from Table 4 that, although loads are generally greater under inbreeding than under random mating, load ratios (L,/Li) greater than unity are possible. For example, with breeding scheme B and loose linkage the load ratio is greater than unity for the mixed under- and overdominance models. Consequently it is expected that inbreeding should improve mean population fitness with certain combinations of selective values when underdominance is one of the features of the genetic model, or whenever y is large and positive. Second, the results suggest that the curvilinearity imposed upon the regression of load on inbreeding by epistasis is, in most instances, probably insufficient to be detected experimentally by regression analysis. In the equation L, = L, -tPI f f P2f”, PI and P2 are, respectively, the linear and quadratic regression coefficients. The nonepistatic models, as expected on the basis of e,, = 0 give p2 = 0. For the epistatic models investigated, PIis generally large. The value of p2is often small except for models of mixed under- overdominance in case of c = .50 but at lower levels of inbreeding this nonlinearity is not likely to be detected experimentally. Thus, while certain models can be developed that involve significant nonlinear

v w w

I I I I I I 02 04 Ob os 10 0 0.2 0.4 0.6 0.8 1.0 INBREEDING COEFFICIENT (I) INBREEDING COEFFICIENT (1)

,FXGURES1 and 2.-The regression of genetic load (Li)on inbreeding (f) . Numbers on right margin refer to genetic models given in the text. Recombination value, c = .01 (Figure 1); c = 50 (Figure 2). Inbreeding was imposed by varying proportions of selfing following one generation of random mating (Scheme B). Note the linear and quadratic components of regression as also givep by the values of PI,P2 (Table 4). EPISTASIS AND GAMETIC UNBALANCE 1037 regression (e.g. KING1966), for the kinds of models studied, load tends to increase approximately linearly with the level of inbreeding. This relationship is depicted graphically in Figures 1 and 2. It is apparent from these figures that the regression coefficient, PI,is strongly correlated with the degree of heterosis and that the primary measure it provides is of the level of heterozygote advantage. This merely confirms what has been known for a long time, that inbreeding is a powerful technique for detecting heterosis. Third, in systems where epistasis and gametic unbalance are important, mating systems which generate equivhlent values of the fixation index (f), may not produce equivalent genetic load. This results from the fact that mating systems which generate an equivalent f generally do not generate an equivalent F,, This can result in different amounts of load generated by sib mating and the mixed selfing and random mating system with s = .50 (Table 4), both of which give f = .25. The relationship between f and F varies from one mating system to another because of the zygotic associations of the type discussed by HALDANE (1949) and BENNETTand BINET(1956). Obviously this effect must be takeii into account should load comparisons be made between different inbreeding systems. Finally, it seems worthwhile to emphasize that for load analyses in experimental populations the realized values off will always be less than expected from various systems of inbreeding when epistasis and/or overdominance maintain an excess of heterozygotes unless one round of random mating without selection can be performed prior to inbreeding (see Table 3).The effect of this bias is to produce a spurious upward curvilinearity in the regression of load on inbreeding similar to that which can be seen in the data of DOBZHANSKY,SPASSKY and TID- WELL (1963), MALOGOLOWKIN-COHENet al. (1964) and SPASSKYet al. (1965) in Drosophila pseudoobscura and D. willistoni. One might imply from this sort of data that in these species there exist epistatic systems in which y (or an analogous parameter for n-locus epistatic systems) is negative and large, and D and (ez2f el,) are of opposite sign. This may well be the case. However, such curvilinearity could easily be an anomaly resulting from regressing load on inappropriate values of f owing to overdominance in the base population; it could also result from the effect of differences in F2, associated with the mating systems utilized to obtain different degrees of inbreeding. The authors wish to thank DRS.J. F. CROWand P. L. WORKMANfor reading the manuscript critically, and the Computer Center at the University of California, Davis, for the use of its facilities.

SUMMARY Genetic load was examined in populations in which epistasis maintains gametic unbalance. Both analytical and Monte Carlo simulation methods were utilized.- An explicit genetic load expression was obtained for %locus cases by utilizing the concept of joint identity by descent for allelic pairs, Pi. This expression shows 1038 P. E. HANSCHE et al. that the regression of load on inbreeding can be curvilinear upward or downward depending on the type of epistasis. The extent of curvilinearity vanes with selective values and decreases with tighter linkage. In general, the curvilinearity is negligible, particularly in the lower range of values of WRIGHT’Sinbreeding coefficient, f. This result suggests that in practice analysis of regression of load on inbreeding is a poor discriminator of epistasis.-The probability of identity by descent for allelic pairs, Fji, depends on the type as well as the amount of inbreeding. The regression of load on inbreeding in epistatic systems consequently also depends on the mating system and when this is not taken into account, a spurious upward curvilinearity may be imposed on the regression of load on inbreeding. A spurious upward curvilinearity can also be imposed on the regression in populations where the maintenance of heterozygotes results in negative values of the fixation index (i.e., where f < 0). Technical problems associated with regressing genetic load on f reinforce the conclusion that regression analysis in systems similar to those studied would often fail to detect the presence of the epistatic component of fitness (or load).

LITERATURE CITED

BENNETT,J. H., and F. E. BINET, 1956 Association between Mendelian factors with mixed selfing and random mating. Heredity 10: 51-55. COCKERHAM,C. C., 1954 An extension of the concept of partitioning hereditary variance for analysis of covariances among relatives when epistasis is present. Genetics 39 : 859-882. CROW,J. F., 1958 Some possibilities for measuring selection intensities in man. Human Biol. 30: 1-13. DOBZHANSKY,TH., B. SPASSKY,and T. TIDWELL,1963 Genetics of natural populations. XXXII. Inbreeding and the mutational and balanced genetic loads in natural populations of Dro- sophila pseudoobscura. Genetics 48: 361-373. HALDANE,J. B. S., 1937 The effect of variation on fitness. Am. Naturalist 71: 337-349. - 194.9 The association of characters as a result of inbreeding and linkage. Ann. Eugenics. 15: 15-23. JAIN, S. K., and R. W. ALLARD,1966 Effects of linkage and epistasis on population changes under inbreeding. Genetics 53 : 633-659. KEMPTHORNE,O., 1957 An Introduction to Genetic Statistics. Wiley, New York. KING, J. L., 1966 The gene interaction component of the genetic load. Genetics 53: 403-413. LEVENE,H., 1963 Inbred genetic loads and the determination of population structure. Proc. Natl. Acad. Sci. U.S. 50: 587-592. MALOGOLOWKIN-COHEN,CH., H. LEVENE,N. P. DOBZHANSKY,and A. S. SIMMONS,1964 Inbreed- ing and the mutational and balanced loads in natural populations of Drosophila willistom.. Genetics 50: 1299-1311. MORTON,N. E., J. F. CROW,and H. J. MULLER,1956 An estimate of the mutational damage in man from data on consanguineous marriages. Proc. Natl. Acad. Sci. U.S. 42: 855-863. NEI, M., 1965 Effect of linkage on the genetic load manifested under inbreeding. Genetics 51: 679-688. EPISTASIS AND GAMETIC UNBALANCE 1039 SCHNELL.F. W., 1961 Some general formulations of linkage effects in inbreeding. Genetics 46: 947-957. SPASSKY,B., TH. DOBZHANSKY,and W. W. ANDERSON,1965 Genetics of natural populations. XXXVI. Epistatic interactions of the components of the genetic load in Drosophila pseudoobscura. Genetics 52: 653-654. WRIGHT,S., 1965 The interpretation of population structure by F-statistics with special regard to systems of mating. Evolution 19 : 395420.

APPENDIX I: Derivation of Ft: when x and y are full sibs. The probability of identity by descent between two gametes g, and g, derived from the full sibs x and y, given that g, is AB, can be expressed as P(g, is AdBd I g, is AB) = Fw. Since g, can arise as a nonrecombinant, e,or recombinant gamete, g:, P(g, is A& 1 g, is AB) = P(gN,R is AB n g, is A&) / P(g,is AB) +P(gt is AB n g, is A&) / P(gxis AB). (1) The first term on the right side of this equation gives the probability that g, is AB, given that alleles A and B in gz were derived from the same parent of x, i.e., g“,“ signifies a nonrecombinant gamete of x while g: indicates a recombinant gamete of x. Taking appropriate parental genotypes into account, we have P (g;”is AB n g, is A&) = 2P (gtnis AB n g, is AdBd I one parent of x or y is AABB)P ( AABB) + 2P(g.:R is AB n g, is A& I one parent is AABb)P(AABb)+ . . . etc. = 2rc’r(%)c’l3g”l+ ~‘r(%~~’l2g2ll1+2{[(%)c’l4(2gllgl?) + [(%>cl‘[(?4)c’12 (2gllglq)} +. . . etc. = r (%I (2d4+ 2C2C’?)(gzll+gllgl?+glIg31) + (%)g11g3d4+ (%)glJg31c”’21 = [ ( ‘/s )gll (2C’,? + 2C2C?) - ( % ) c2C‘“Dl (2) Note that for D = 0, expression (2) reduces to that of HALDANE(1949). The second term on the right side of expression (1) is the probability that g, is AB, given that alleles A and B of g, were derived from recombination between parental genotypes of x,that is, P(gT is AB n g, is A&) = P(g, is A&lgr is recombinant n g;. is AB)P(g,is recombinant n g, is AB). (3) Now, since P(gzis AB) = gll - CD and P ( gs is nonrecombinant f‘ g, is AB) = c’g’ll + (%)C’(2gllgl3)+ (i/)~’c(2g~lg31)~(%)c’(2gllg33)=gllc’, we have, P(gz is recombinant n g, is AB) = g,,-cD-g,,c’ = c(gll-D). The probability that y has alleles A and B derived from two parents carrying these alleles respectively is %, so that P (g, is AB I y has alleles A and B derived from two parents) = [(%)cl (?A) and therefore, (3) becomes P(g; is AB n g, is A&) = ( l/)c2 (gll - 0). (4) 1OLEO P. E. HANSCHE et a2. From (2) and (4),we finally have P(gvis AdBd I g, is AB)=[ ( l/)gll(2~”+2c~c‘~+c~) - ( $4 ) c2D X [ ( ?4) + ( $4 ~’’1I / (gli-cD ) and Ft: = P(gvis AdBd 1 gx is AB)P(g, is AB) = { (1/8)gll(W4+2c2d2+c2) - ($4)c2D[(%I -t (1/)d21). (5) Expression (5) reduces to HALDANE’Sexpression (1/8) (2d4+2Czd2+C2)for the case D = 0 where Fx, = gllF$ + g,,Ft: -t g,,F$ f g33F:;. For D # 0, analogous expressions for F$ FZ, F;: need also be derived in the same way as FZ given here.